$3)(25$ points$)$ Suppose that the production function of a firm $-$ where $\backslash ($ K $\backslash )$ is fixed, $\backslash ($ L $\backslash )$ is variable $-$ exhibits first increasing marginal returns, and then diminishing marginal returns.

a$)$ Illustrate the graphs of the production function, cost function and the marginal cost function. Explain.

b$)$ Write the profit maximization of the firm where the firm chooses the optimal amount of production $($output$),$ given the price of the output and the cost function. Characterize the optimal point using calculus $($necessary and sufficient conditions$).$ What is the interpretation of the optimality condition?

c$)$ Show the optimal point graphically, using the optimality condition $($in part $[$b$]).$ What is the effect of an increase in the wage rate on the output decision?

d$)$ Now, suppose that K is also variable. What is the effect of an increase in the wage rate on the optimal K and L decision within the profit$-$maximization problem. Note that the firm can adjust its output decision!